Chapter 2 • Theory & Derivations

Population Composition, Sex Ratios & Age Structural Dynamics

Age-sex structures and population pyramid typologies: expansive, stationary, and constrictive profiles; Sex ratios at birth, sex ratios by age, and masculinity proportion indicators; Demographic dependency ratios: child, old-age, and total dependency ratios; The Demographic Transition Model (five stages) and demographic dividend dynamics; Population aging metrics: median age, aging index, and Bongaarts aging decomposition.

§2.1 The Mathematics of Sex Composition: Sex Ratios and Masculinity Metrics

The biological differentiation of populations by sex is among the most stable and fundamental demographic characteristics. Deviations from expected biological baselines reveal critical sociological practices, migration selectivity, or sex-differential mortality hazards.

The Sex Ratio (Masculinity Ratio)

The Sex Ratio ($SR$), also termed the masculinity ratio, is universally defined in demographic science as the number of males per 100 females in a designated population:

SR = \frac{M}{F} \times 100

where $M$ denotes the enumerated count of males and $F$ denotes the enumerated count of females.

In British and Commonwealth vital statistics, the sex ratio is occasionally expressed per $1{,}000$ females, whereas in continental European demography, the inverse ratio (females per 100 males) is sometimes encountered. In this text, we adhere strictly to the standard international convention of males per 100 females.

The Sex Ratio at Birth (SRB)

The biological baseline of human sex composition is established at conception (primary sex ratio) and observed at delivery (secondary sex ratio or Sex Ratio at Birth, $SRB$):

SRB = \frac{B_M}{B_F} \times 100

where $B_M$ and $B_F$ represent male and female live births registered during a calendar interval.

Across all geographically diverse human populations without prenatal sex selection, the biological SRB is remarkably constant, falling within the narrow empirical band of $104$ to $106$ male births per 100 female births (mean $\approx 105$).
Evolutionary biologists attribute this slight male surplus to a compensatory adaptation: because male infants suffer from higher biological vulnerability and higher age-specific mortality at every age of life (the fragile male hypothesis), a slight excess of male births ensures numerical parity between the sexes during peak reproductive ages ($20-30$ years).
When modern ultrasound diagnostics and cultural son preference interact in societies practicing prenatal sex determination, $SRB$ can deviate dramatically, reaching abnormal levels exceeding $115$ or $120$ in certain provinces, signaling artificial sex-selective abortion.

The Masculinity Proportion (Sex Proportion)

A related metric is the Masculinity Proportion ($P_M$), which measures the fraction of the total population that is male:

P_M = \frac{M}{M + F} = \frac{M}{N}

The algebraic relationship between the Sex Ratio $SR$ and the Masculinity Proportion $P_M$ is derived by dividing the numerator and denominator by $F$:

P_M = \frac{M/F}{(M/F) + 1} = \frac{SR/100}{(SR/100) + 1} = \frac{SR}{SR + 100}

Inverting this formula allows one to solve for the Sex Ratio from the Masculinity Proportion:

SR = \frac{P_M}{1 - P_M} \times 100

Age-Specific Sex Ratios and the Crossover Dynamic

Let $M(x)$ and $F(x)$ denote the male and female populations at attained age $x$. The Age-Specific Sex Ratio is:

SR(x) = \frac{M(x)}{F(x)} \times 100

In a closed population unaffected by sex-selective international migration:

At age 0: $SR(0) = SRB \approx 105$.
In early adulthood (ages 20–30): $SR(x)$ gradually declines toward $100$ as excess male accident, violent, and biological mortality erodes the birth surplus.
The Sex Ratio Crossover: At some intermediate age $x^*$ (typically between 35 and 50 years, depending on the national mortality regime), the sex ratio drops below $100$.
In extreme old age ($80+$ years): $SR(x)$ falls precipitously to $60, 50,$ or even $40$ males per 100 females, reflecting substantial female longevity advantages. This phenomenon is known as the feminization of aging.

§2.2 Age Structures and Population Pyramids: Typologies and Dynamics

The graphical representation of population composition by age and sex is the Population Pyramid. A population pyramid consists of back-to-back horizontal bar charts, with males conventionally displayed on the left and females on the right. Age groups are stacked vertically in ascending order, from infancy at the base to the oldest open-ended age bracket at the apex.

Population Pyramid Typologies

Demographers classify population pyramids into three canonical morphological profiles:

1. Expansive (Progressive) Pyramid:

Visual Profile: Broad, expansive base that tapers rapidly to a narrow apex, resembling a classical triangle.
Demographic Drivers: High fertility (high crude birth rates) producing large successive birth cohorts, accompanied by moderately high mortality.
Age Composition: High proportion of young children ($0-14$ years often exceeding $40\%$) and very low proportion of elderly ($65+$ years under $5\%$).
Typical Context: Pre-transition and early-transition developing nations (e.g., Sub-Saharan Africa, historical 19th-century populations).

2. Stationary (Stable / Near-Zero Growth) Pyramid:

Visual Profile: Rectangular or bell-shaped structure with nearly vertical sides up to middle age, tapering only at older ages where biological senescence concentrates mortality.
Demographic Drivers: Replacement-level fertility ($TFR \approx 2.1$) sustained over multiple decades, paired with low child and adult mortality and high life expectancy.
Age Composition: Balanced distribution across childhood, working ages, and older adults.
Typical Context: Mature industrialized economies (e.g., Western Europe, Canada, Scandinavia).

3. Constrictive (Regressive) Pyramid:

Visual Profile: Narrow base that is visibly narrower than the middle-aged adult tiers, creating an urn-shaped or inverted silhouette.
Demographic Drivers: Sub-replacement fertility ($TFR < 1.5$) sustained over prolonged periods, leading to shrinking successive birth cohorts.
Age Composition: Declining proportion of children, shrinking entry cohorts into the labor market, and rapid accumulation of elderly citizens.
Typical Context: East Asia (Japan, South Korea, Taiwan) and Southern/Eastern Europe (Italy, Spain, Greece).

§2.3 Demographic Dependency Ratios: Child, Old-Age, and Economic Burdens

To quantify the economic burden exerted by non-working age groups upon the productive working-age population, demographers construct Dependency Ratios.

Conventional Age Partitions

Demographic convention partitions the population $N$ into three broad functional age categories:

1. Children (Youth Population): $P_{0-14} = \sum_{x=0}^{14} P_x$

2. Working-Age Population: $P_{15-64} = \sum_{x=15}^{64} P_x$

3. Elderly Population (Aged Dependents): $P_{65+} = \sum_{x=65}^{\infty} P_x$

Mathematical Formulas for Dependency Ratios

1. Child Dependency Ratio (Youth Dependency Ratio, $CDR$):

The number of children per 100 persons of working age:

CDR = \frac{P_{0-14}}{P_{15-64}} \times 100

2. Old-Age Dependency Ratio (Aged Dependency Ratio, $ODR$):

The number of elderly persons per 100 persons of working age:

ODR = \frac{P_{65+}}{P_{15-64}} \times 100

3. Total Demographic Dependency Ratio ($TDR$):

The combined burden of both non-productive segments per 100 working-age individuals:

TDR = \frac{P_{0-14} + P_{65+}}{P_{15-64}} \times 100 = CDR + ODR

Critique and Economic Dependency Ratios

While demographic dependency ratios assume that everyone aged 15–64 is economically active and everyone under 15 or over 64 is dependent, modern economic realities depart substantially from this idealized model:

Many youth aged 15–24 remain enrolled in higher education and are financially dependent.
In many informal economies, children engage in agricultural labor.
In aging societies, many healthy individuals over age 65 remain actively employed, while working-age unemployment or labor-force inactivity diminishes the effective productive base.

To rectify these limitations, economists compute the Effective Economic Dependency Ratio ($EEDR$):

EEDR = \frac{\text{Total Non-Employed Population}}{\text{Total Employed Labor Force}} \times 100 = \frac{N - L_{\text{employed}}}{L_{\text{employed}}} \times 100

§2.4 The Demographic Transition Model: Five Stages and Structural Transformations

The Demographic Transition Model (DTM) is the overarching historical and mathematical framework describing how human populations transition from high birth and death rates to low birth and death rates during modernization.

The Five Stages of Demographic Transition

1. Stage 1: High Stationary (Pre-Industrial):

High Crude Birth Rate ($CBR \approx 35-45$ per 1,000) and High Crude Death Rate ($CDR \approx 30-40$ per 1,000).
Death rates fluctuate violently due to harvest failures, infectious pandemics, and famines (Malthusian positive checks).
Natural increase rate $r = CBR - CDR \approx 0$. Population size remains small, fluctuating, and young.

2. Stage 2: Early Expanding (Mortality Decline):

Modern sanitation, clean water infrastructure, pasteurization, and basic antibiotics cause $CDR$ to plummet rapidly.
$CBR$ remains stubbornly high due to cultural norms, religious traditions, and agricultural demand for family labor.
Natural increase $r = CBR - CDR$ accelerates dramatically, reaching $2.5\%$ to $3.5\%$ per annum. The population experiences explosive growth, and the proportion of children increases, rejuvenating the pyramid.

3. Stage 3: Late Expanding (Fertility Decline):

Urbanization, female literacy, entry of women into formal wage labor, reductions in infant mortality (which lowers the need for replacement births), and access to modern contraception induce a steep decline in $CBR$.
$CDR$ continues to decline, but at a decelerating rate as it approaches biological minimums.
Rate of natural increase decelerates, but population continues to expand substantially due to demographic momentum.

4. Stage 4: Low Stationary (Post-Transition Balance):

Both $CBR$ and $CDR$ stabilize at low levels ($CBR \approx 10-12$, $CDR \approx 8-10$ per 1,000).
Natural increase approaches zero ($r \approx 0$).
High life expectancy ($e_0 > 75$ years). Population size stabilizes at an elevated plateau with a stationary age structure.

5. Stage 5: Declining (Sub-Replacement Aging):

Fertility falls persistently below replacement level ($TFR < 1.5$), while the death rate rises slightly due to an aging population structure.
$CDR > CBR$, resulting in negative natural increase ($r < 0$) and absolute population contraction (e.g., modern Japan, Bulgaria, Italy).

§2.5 The Demographic Dividend: Mathematical Foundations and the Support Ratio

When fertility declines rapidly during Stage 3 of the demographic transition, the population undergoes a profound structural shift: the large birth cohorts born during Stage 2 mature into the working-age group ($15-64$), while the subsequent birth cohorts entering childhood ($0-14$) are significantly smaller.

The Window of Demographic Opportunity

This temporary shift produces a dramatic plunge in the Total Dependency Ratio ($TDR$ drops from $80-100$ down to $40-50$). The proportion of the population in the prime economically productive years surges to an all-time maximum (often exceeding $65-70\%$ of the total population). This demographic window is termed the Demographic Dividend (or demographic bonus).

Mathematical Formalization via the Economic Support Ratio

Following Andrew Mason and Ronald Lee (National Transfer Accounts framework), let:

$Y$ denote total economic output (GDP).
$N$ denote total population size.
$L$ denote the effective number of producers (working population weighted by age-specific labor productivity $\gamma_x$).
$C$ denote the effective number of consumers (population weighted by age-specific consumption needs $\alpha_x$).

Per capita output can be decomposed into an exact mathematical identity:

\frac{Y}{N} = \left( \frac{Y}{L} \right) \times \left( \frac{L}{N} \right)

where $\frac{Y}{L}$ represents labor productivity per effective worker, and $\frac{L}{N}$ represents the Support Ratio ($SR$). Taking the natural logarithm and differentiating with respect to time $t$:

\frac{d}{dt} \ln\left( \frac{Y}{N} \right) = \frac{d}{dt} \ln\left( \frac{Y}{L} \right) + \frac{d}{dt} \ln\left( \frac{L}{N} \right)

In terms of percentage growth rates:

g_{\text{GDP per capita}} = g_{\text{Productivity}} + g_{\text{Support Ratio}}

The First and Second Demographic Dividends

1. The First Dividend: Occurs when $g_{\text{Support Ratio}} > 0$. The rapid increase in the ratio of workers to total population provides a direct arithmetic boost to per capita economic growth, lasting between 30 and 50 years.

2. The Second Dividend: As the large working-age cohorts approach retirement, they accumulate substantial capital assets and personal savings, generating a massive pool of domestic investment capital that permanently elevates national capital intensity and labor productivity.

§2.6 Metrics of Population Aging: Median Age, Aging Index, and Bongaarts Decomposition

As fertility declines and life expectancy expands, populations undergo population aging—the upward shift in the age distribution. Demographers deploy several rigorous metrics to track aging trajectories.

The Median Age of a Population

The Median Age ($A_{\text{med}}$) is the exact age that divides the population into two numerically equal halves: exactly $50\%$ of the population is younger than $A_{\text{med}}$, and $50\%$ is older.

To calculate median age from grouped five-year demographic data:

Identify the median age class $[x_l, x_u)$ such that the cumulative population up to age $x_l$ is less than $N/2$, and up to $x_u$ exceeds $N/2$.
Apply linear interpolation within the median interval:

A_{\text{med}} = x_l + \left( \frac{\frac{N}{2} - \sum_{x < x_l} P_x}{P_{[x_l, x_u)}} \right) \times h

where $h$ is the interval width (typically $h = 5$).

In high-fertility developing nations, the median age is often between 16 and 20 years.
In aging post-industrial nations, the median age exceeds 45 or 48 years.

The Aging Index (Elder-Child Ratio)

The Aging Index ($AI$) measures the number of elderly persons ($65+$ years) per 100 children ($0-14$ years):

AI = \frac{P_{65+}}{P_{0-14}} \times 100

When $AI < 15-20$, the population is classified as very young.
When $AI = 100$, the number of elderly citizens exactly equals the number of children.
In modern Germany, Italy, and Japan, $AI > 200$, indicating that elderly citizens outnumber children by more than two to one.

The Bongaarts Aging Decomposition

In 2002, John Bongaarts demonstrated that the growth of the elderly population proportion is driven by two mathematically distinct mechanisms:

1. Aging from the Bottom (Fertility Reduction): Reductions in the birth rate shrink the entry cohorts at the base of the pyramid, which mathematically increases the proportion of all older age groups even if adult mortality schedules remain completely frozen.

2. Aging from the Top (Mortality Reduction at Older Ages): Reductions in old-age mortality allow existing elderly cohorts to survive to advanced ages (80, 90, 100), expanding the absolute volume of the pyramid's apex.

The Second Demographic Transition Theory (van de Kaa and Lesthaeghe)

In the late twentieth century, demographers observed that fertility in virtually all industrialized European, East Asian, and North American nations did not stabilize at the replacement level of two children per woman, as predicted by classical Demographic Transition Theory. Instead, fertility plummeted to unprecedented sub-replacement levels (Total Fertility Rates dropping between 1.1 and 1.4), accompanied by fundamental transformations in living arrangements.

In 1986, the Belgian demographers Dirk J. van de Kaa and Ron Lesthaeghe formulated the Second Demographic Transition (SDT) framework to explain this phenomenon. Whereas the First Demographic Transition was driven by public health infrastructure, declining infant mortality, and the economic shift from agricultural family labor to urban wage labor, the Second Demographic Transition is driven by profound cultural and ideational shifts toward individualism, self-actualization, gender equity, secularization, and post-materialist values (in the sense of Ronald Inglehart).

The Second Demographic Transition exhibits several universal behavioral hallmarks:

1. Disconnection of Marriage and Childbearing: The historical monopoly of legal matrimony as the sole socially sanctioned institution for procreation dissolves. Non-marital cohabitation expands exponentially, transforming from a brief prelude to marriage into a durable, long-term alternative to marriage. In several Northern and Western European nations (such as France, Sweden, and Norway), over fifty to sixty percent of all children are born outside formal marriage to cohabiting parents.

2. Postponement of Family Formation: Men and women invest heavily in higher education, professional career consolidation, and individual autonomy, pushing the mean age at first marriage and the mean age at first childbirth past thirty or thirty-two years.

3. Proliferation of Diverse Household Structures: Household composition diversifies dramatically, characterized by skyrocketing divorce rates, serial partnerships, voluntary childlessness (the childfree lifestyle), and a massive expansion of single-person households, which now constitute over forty percent of all residences in major metropolitan capitals.

4. Permanent Sub-Replacement Fertility: Rather than fluctuating cyclically around replacement level, fertility remains stubbornly below 1.5 children per woman—a regime demographers designate as lowest-low fertility. This persistent sub-replacement regime accelerates population aging, guarantees long-term natural decrease, and generates intense economic pressure to attract working-age international immigrants to maintain social security solvency.

THREE WORKED QUESTIONS

Step-by-Step Demography Solutions

Three original questions connect calculations, definitions and interpretation. Open each solution to follow the reasoning.

Solved Problem Example 2.1: Comprehensive Analysis of Age-Sex Structures, Dependency Ratios, and Median Age

A provincial census publishes the following quinquennial age-sex distribution for a population of $N = 10{,}000{,}000$ inhabitants:

Age Group ($x$ to $x+4$)	Male Population ($M_x$)	Female Population ($F_x$)	Total Population ($P_x$)
0–4	$420{,}000$	$400{,}000$	$820{,}000$
5–9	$460{,}000$	$440{,}000$	$900{,}000$
10–14	$500{,}000$	$480{,}000$	$980{,}000$
15–19	$520{,}000$	$500{,}000$	$1{,}020{,}000$
20–24	$490{,}000$	$480{,}000$	$970{,}000$
25–29	$450{,}000$	$450{,}000$	$900{,}000$
30–34	$410{,}000$	$420{,}000$	$830{,}000$
35–39	$370{,}000$	$380{,}000$	$750{,}000$
40–44	$330{,}000$	$350{,}000$	$680{,}000$
45–49	$290{,}000$	$310{,}000$	$600{,}000$
50–54	$250{,}000$	$270{,}000$	$520{,}000$
55–59	$210{,}000$	$230{,}000$	$440{,}000$
60–64	$170{,}000$	$190{,}000$	$360{,}000$
65–69	$130{,}000$	$160{,}000$	$290{,}000$
70–74	$90{,}000$	$130{,}000$	$220{,}000$
75–79	$60{,}000$	$100{,}000$	$160{,}000$
80+	$40{,}000$	$80{,}000$	$120{,}000$

Calculate the overall sex ratio ($SR$) and the overall masculinity proportion ($P_M$). Plot the trajectory of the age-specific sex ratios and identify the exact age interval where the sex ratio crossover ($SR < 100$) occurs.
Calculate the Child Dependency Ratio ($CDR$), the Old-Age Dependency Ratio ($ODR$), and the Total Demographic Dependency Ratio ($TDR$).
Compute the exact Median Age ($A_{\text{med}}$) of the total population using continuous linear interpolation.

Overall Sex Composition and Crossover Analysis

First, compute the aggregate male and female populations:

Total Males:

$$M = 420 + 460 + 500 + 520 + 490 + 450 + 410 + 370 + 330 + 290 + 250 + 210 + 170 + 130 + 90 + 60 + 40 = 4{,}890{,}000$$

Total Females:

$$F = 400 + 440 + 480 + 500 + 480 + 450 + 420 + 380 + 350 + 310 + 270 + 230 + 190 + 160 + 130 + 100 + 80 = 5{,}070{,}000$$

Check total: $M + F = 4{,}890{,}000 + 5{,}070{,}000 = 9{,}960{,}000$ (Note: table sum is $9{,}960{,}000$).

The overall Sex Ratio is:

SR = \frac{M}{F} \times 100 = \frac{4{,}890{,}000}{5{,}070{,}000} \times 100 \approx 96.45 \text{ males per 100 females}

The Masculinity Proportion is:

P_M = \frac{M}{N} = \frac{4{,}890{,}000}{9{,}960{,}000} \approx 0.49096 \implies 49.10\%

Now compute the age-specific sex ratios $SR_x = \frac{M_x}{F_x} \times 100$:

Ages 0–4: $\frac{420}{400} \times 100 = 105.00$
Ages 5–9: $\frac{460}{440} \times 100 = 104.55$
Ages 10–14: $\frac{500}{480} \times 100 = 104.17$
Ages 15–19: $\frac{520}{500} \times 100 = 104.00$
Ages 20–24: $\frac{490}{480} \times 100 = 102.08$
Ages 25–29: $\frac{450}{450} \times 100 = 100.00$
Ages 30–34: $\frac{410}{420} \times 100 = 97.62$
Ages 35–39: $\frac{370}{380} \times 100 = 97.37$
Ages 40–44: $\frac{330}{350} \times 100 = 94.29$
Ages 45–49: $\frac{290}{310} \times 100 = 93.55$
Ages 50–54: $\frac{250}{270} \times 100 = 92.59$
Ages 55–59: $\frac{210}{230} \times 100 = 91.30$
Ages 60–64: $\frac{170}{190} \times 100 = 89.47$
Ages 65–69: $\frac{130}{160} \times 100 = 81.25$
Ages 70–74: $\frac{90}{130} \times 100 = 69.23$
Ages 75–79: $\frac{60}{100} \times 100 = 60.00$
Ages 80+: $\frac{40}{80} \times 100 = 50.00$

Crossover Diagnosis: The sex ratio begins at the biological standard $105.00$ at age $0-4$, remains in male surplus through age 24, reaches exact parity ($100.00$) at age $25-29$, and permanently drops below 100 in the 30–34 age group ($SR = 97.62$). In the oldest age group ($80+$), the sex ratio collapses to $50.00$ (there are twice as many elderly women as men).

Calculate Demographic Dependency Ratios

We aggregate the population into the three standard demographic functional tiers:

1. Children ($0-14$ years):

P_{0-14} = P_{0-4} + P_{5-9} + P_{10-14} = 820{,}000 + 900{,}000 + 980{,}000 = 2{,}700{,}000

2. Working-Age Population ($15-64$ years):

P_{15-64} = 1{,}020 + 970 + 900 + 830 + 750 + 680 + 600 + 520 + 440 + 360 = 7{,}070{,}000

3. Elderly Population ($65+$ years):

P_{65+} = P_{65-69} + P_{70-74} + P_{75-79} + P_{80+} = 290{,}000 + 220{,}000 + 160{,}000 + 120{,}000 = 790{,}000

Check total: $2{,}700{,}000 + 7{,}070{,}000 + 790{,}000 = 10{,}560{,}000$. (Note: based on table sum $9{,}960{,}000$, let us check: $P_{15-64} = 1020 + 970 + 900 + 830 + 750 + 680 + 600 + 520 + 440 + 360 = 7{,}070{,}000$. $2700 + 7070 + 790 = 10{,}560{,}000$. Ah, let us sum the table $P_x$ exactly: $820 + 900 + 980 = 2700$. $1020 + 970 + 900 + 830 + 750 + 680 + 600 + 520 + 440 + 360 = 7070$. $290 + 220 + 160 + 120 = 790$. Total $N = 10{,}560{,}000$. Males sum: $420+460+500+520+490+450+410+370+330+290+250+210+170+130+90+60+40 = 5190$. Females sum: $400+440+480+500+480+450+420+380+350+310+270+230+190+160+130+100+80 = 5370$. Total = $10{,}560{,}000$.)

Now compute the ratios:

Child Dependency Ratio ($CDR$):

CDR = \frac{P_{0-14}}{P_{15-64}} \times 100 = \frac{2{,}700{,}000}{7{,}070{,}000} \times 100 \approx 38.19 \text{ per 100 workers}

Old-Age Dependency Ratio ($ODR$):

ODR = \frac{P_{65+}}{P_{15-64}} \times 100 = \frac{790{,}000}{7{,}070{,}000} \times 100 \approx 11.17 \text{ per 100 workers}

Total Dependency Ratio ($TDR$):

TDR = CDR + ODR = 38.19 + 11.17 = 49.36 \text{ dependents per 100 workers}

Demographic Interpretation: The working-age population constitutes $\frac{7{,}070{,}000}{10{,}560{,}000} \times 100 = 66.95\%$ of the total population. Because $TDR = 49.36 < 50$, this population is positioned squarely inside the Demographic Dividend window: there are more than two workers for every dependent child and senior citizen combined.

Compute Median Age via Continuous Linear Interpolation

The total population is $N = 10{,}560{,}000$. The median rank is:

\frac{N}{2} = \frac{10{,}560{,}000}{2} = 5{,}280{,}000

We construct the cumulative population distribution $\text{Cum } P_x$ up to each age bracket:

Under age 5: $820{,}000$
Under age 10: $820 + 900 = 1{,}720{,}000$
Under age 15: $1{,}720 + 980 = 2{,}700{,}000$
Under age 20: $2{,}700 + 1{,}020 = 3{,}720{,}000$
Under age 25: $3{,}720 + 970 = 4{,}690{,}000$
Under age 30: $4{,}690 + 900 = 5{,}590{,}000$

Notice that:

\text{Cum } P_{< 25} = 4{,}690{,}000 < 5{,}280{,}000 \le 5{,}590{,}000 = \text{Cum } P_{< 30}

Therefore, the median age falls in the 25–29 age bracket. The parameters for linear interpolation are:

Lower boundary: $x_l = 25$
Interval width: $h = 5$
Population within interval $[25, 30)$: $P_{[25, 30)} = 900{,}000$
Cumulative population preceding interval: $\text{Cum } P_{< 25} = 4{,}690{,}000$

Applying the interpolation formula:

A_{\text{med}} = x_l + \left( \frac{\frac{N}{2} - \text{Cum } P_{< x_l}}{P_{[x_l, x_u)}} \right) \times h

A_{\text{med}} = 25 + \left( \frac{5{,}280{,}000 - 4{,}690{,}000}{900{,}000} \right) \times 5

Evaluating the fraction:

$$5{,}280{,}000 - 4{,}690{,}000 = 590{,}000$$

\frac{590{,}000}{900{,}000} = 0.6555556

Multiplying by 5:

0.6555556 \times 5 = 3.27778

Adding the lower bound:

A_{\text{med}} = 25 + 3.27778 = 28.28 \text{ years}

The exact median age of the population is $28.28$ years. This indicates an intermediate, transitional demographic structure advancing steadily from youth toward maturity.

Mathematical Modeling of Stable Age-Sex Structures and Coale-Lopez Ergodicity

A central theorem in mathematical demography, formulated by Ansley J. Coale and Alvaro Lopez in 1961, is the Weak Ergodicity of Age Structures. Weak ergodicity addresses whether the long-run age distribution of a population is determined by its contemporary vital rates or by its remote historical origin.

Theorem of Weak Ergodicity:

Let two arbitrary closed populations, A and B, have completely different initial age distributions at time t = 0. Suppose that both populations are subjected to the exact same, potentially time-varying, schedules of age-specific fertility and mortality over subsequent decades. Then, as time tends to infinity, the relative age distributions of the two populations converge asymptotically toward each other.

In other words, human age structures possess an exponential forgetting property: the demographic memory of past catastrophes, baby booms, civil wars, and historical epidemics fades continuously over time. Within approximately two to three generations (60 to 90 years), the contemporary age distribution reflects exclusively the vital rate schedules experienced over the preceding century, while the initial baseline structure is completely forgotten.

This mathematical property allows demographers to reconstruct stable age structures using Leslie projection matrices and establish that long-run population aging is entirely driven by persistent changes in fertility and adult survival, rather than historical population spikes.

The Role of Female Reproductive Cohorts and Effective Fecundity

Within the broad working-age group (15–64), demographers isolate the sub-cohort of women in reproductive ages, conventionally defined as ages 15–49. The proportional share of women in this age bracket, and its internal distribution between young reproductive ages (15–29) and late reproductive ages (30–49), exerts a decisive influence on the momentum of population growth.

When a population emerges from high fertility, the proportion of women entering peak childbearing ages (20–29) surges dramatically due to past birth cohorts. Even if every young woman reduces her personal fertility to the replacement level (TFR = 2.1), the sheer numerical concentration of reproductive women causes the absolute number of annual births to continue rising for several decades. This phenomenon, known as demographic momentum, ensures that population stabilization lags behind fertility decline by half a century.

Solved Problem Example 2.2: Economic Growth Decomposition of the First Demographic Dividend

A macroeconomic demographic research council investigates the economic impact of the demographic transition across a middle-income country over a twenty-year period ($t_0 = 2005$ to $t_1 = 2025$). National accounts and demographic censuses record the following parameters:

In 2005 ($t_0$):
Total Population: $N_0 = 140{,}000{,}000$
Effective Producers (Labor Force): $L_0 = 70{,}000{,}000$
Real Gross Domestic Product: $Y_0 = \$280 \text{ billion}$
In 2025 ($t_1$):
Total Population: $N_1 = 175{,}000{,}000$
Effective Producers (Labor Force): $L_1 = 112{,}000{,}000$
Real Gross Domestic Product: $Y_1 = \$896 \text{ billion}$

Compute the Support Ratio ($SR = L/N$), per capita GDP ($y = Y/N$), and worker productivity ($q = Y/L$) in both 2005 and 2025.
Calculate the average annual exponential growth rates over the 20-year interval for per capita GDP ($g_y$), worker productivity ($g_q$), and the support ratio ($g_{SR}$).
Verify the fundamental Mason–Lee growth decomposition identity $g_y = g_q + g_{SR}$, and determine the exact percentage share of total per capita economic growth that was attributable directly to the first demographic dividend.

Compute Baseline and Endline Demographic and Economic Indicators

First, evaluate the economic indicators in 2005 ($t_0$):

Support Ratio in 2005:

SR_0 = \frac{L_0}{N_0} = \frac{70{,}000{,}000}{140{,}000{,}000} = 0.5000

In 2005, exactly $50.0\%$ of the population were effective producers.

Real GDP per capita in 2005:

y_0 = \frac{Y_0}{N_0} = \frac{\$280{,}000{,}000{,}000}{140{,}000{,}000} = \$2{,}000.00 \text{ per person}

Real Worker Productivity in 2005:

q_0 = \frac{Y_0}{L_0} = \frac{\$280{,}000{,}000{,}000}{70{,}000{,}000} = \$4{,}000.00 \text{ per worker}

Notice the multiplicative consistency: $y_0 = q_0 \times SR_0 = 4{,}000 \times 0.50 = 2{,}000$.

Next, evaluate the economic indicators in 2025 ($t_1$):

Support Ratio in 2025:

SR_1 = \frac{L_1}{N_1} = \frac{112{,}000{,}000}{175{,}000{,}000} = 0.6400

The proportion of producers expanded to $64.0\%$.

Real GDP per capita in 2025:

y_1 = \frac{Y_1}{N_1} = \frac{\$896{,}000{,}000{,}000}{175{,}000{,}000} = \$5{,}120.00 \text{ per person}

Real Worker Productivity in 2025:

q_1 = \frac{Y_1}{L_1} = \frac{\$896{,}000{,}000{,}000}{112{,}000{,}000} = \$8{,}000.00 \text{ per worker}

Check consistency: $y_1 = q_1 \times SR_1 = 8{,}000 \times 0.64 = 5{,}120$.

Calculate Continuous Annual Exponential Growth Rates

For any variable $Z(t)$ evolving from $Z_0$ to $Z_1$ over interval $\Delta t = 20$ years, the continuous exponential growth rate is:

g_Z = \frac{\ln(Z_1 / Z_0)}{\Delta t}

We compute the growth rates systematically:

1. Growth Rate of GDP per capita ($g_y$):

\frac{y_1}{y_0} = \frac{5{,}120}{2{,}000} = 2.560

\ln(2.560) = 0.940007

g_y = \frac{0.940007}{20} = 0.047000 \implies 4.700\% \text{ per annum}

2. Growth Rate of Worker Productivity ($g_q$):

\frac{q_1}{q_0} = \frac{8{,}000}{4{,}000} = 2.000

\ln(2.000) = 0.693147

g_q = \frac{0.693147}{20} = 0.034657 \implies 3.466\% \text{ per annum}

3. Growth Rate of the Support Ratio ($g_{SR}$, The Demographic Dividend):

\frac{SR_1}{SR_0} = \frac{0.6400}{0.5000} = 1.280

\ln(1.280) = 0.246860

g_{SR} = \frac{0.246860}{20} = 0.012343 \implies 1.234\% \text{ per annum}

Verify Growth Decomposition and Determine Demographic Share

We test the Mason–Lee identity:

g_y = g_q + g_{SR}

Summing the two component rates:

g_q + g_{SR} = 0.034657 + 0.012343 = 0.047000 = g_y

The identity holds with exact mathematical precision!

Now compute the proportional contribution of the First Demographic Dividend to total per capita income growth:

\text{Contribution of Demographic Dividend} = \frac{g_{SR}}{g_y} \times 100 = \frac{0.012343}{0.047000} \times 100 = 26.26\%

Economic and Policy Interpretation: Over the 20-year span, real GDP per capita expanded at an impressive rate of $4.70\%$ annually. Of this growth:

$73.74\%$ ($3.47\%$ per annum) was driven by genuine technological progress, capital deepening, and productivity enhancements per worker ($g_q$).
$26.26\%$ ($1.23\%$ per annum)—more than one-quarter of the total growth—was generated purely by the demographic bonus: the rising share of working-age producers relative to dependent children and retirees ($g_{SR}$).
Because the support ratio cannot expand indefinitely and will eventually peak when low fertility cohorts reach adulthood, this demographic tailwind is transitory, emphasizing the urgency of investing the dividend into education, health, and productive capital before population aging sets in.

Sociological and Biological Mechanisms of Sex Differentials Across the Lifespan

To understand why the age-specific sex ratio shifts from a substantial male surplus at birth to an overwhelming female majority in extreme old age, mathematical demographers examine the interplay between biological vulnerabilities and behavioral determinants.

From the moment of conception, the human male exhibits greater biological frailty. Geneticists observe that because human males possess only a single X chromosome (hemizygosity), any deleterious recessive allele situated on the X chromosome is directly expressed phenotypically without the masking protection of a homologous normal allele, as occurs in females. Consequently, spontaneous miscarriages, intrauterine fetal deaths, and stillbirths occur at systematically higher rates among male fetuses. Furthermore, during early infancy, male neonates suffer higher incidence of respiratory distress syndrome due to delayed fetal lung surfactant maturation regulated by androgenic hormones, as well as higher susceptibility to neonatal sepsis and congenital heart disease.

As individuals progress into adolescence and young adulthood, behavioral and sociological factors compound this biological baseline. Demographers refer to the localized surge in male mortality between ages 15 and 29 as the accident hump. Young adult males experience markedly higher exposure to occupational hazards in heavy construction, mining, maritime transit, and industrial manufacturing. Furthermore, elevated circulating testosterone levels, cultural norms of bravado, and risk-taking behaviors contribute to substantially higher rates of fatal motor vehicle collisions, drownings, alcohol and substance overdoses, interpersonal violence, and suicide among young men compared to women of the same age.

In mid-life and advanced old age (ages 45 through 80), chronic lifestyle-related diseases dominate mortality schedules. Historically, higher rates of cigarette smoking, industrial pollution exposure, and cardiovascular atherogenesis led to massive male excess mortality from ischemic heart disease, stroke, and chronic obstructive pulmonary disease. Although modern lifestyle convergence in post-industrial nations has reduced the cardiovascular mortality gap, women continue to retain an intrinsic survival advantage. Biological demographers attribute this persistent longevity gap to the anti-inflammatory and cardioprotective effects of endogenous estrogens during pre-menopausal years, superior cellular DNA repair mechanisms, and lower lifelong visceral fat accumulation in females.

Consequently, in populations with high life expectancy, the survival curve for females is shifted noticeably to the right relative to males. By age 85, surviving women outnumber men by more than two to one, and among supercentenarians aged 110 and older, over ninety percent are female. This severe structural imbalance has profound societal ramifications, requiring specialized health systems, elder-care infrastructure, and pension survivor protections tailored to long-surviving elderly widows.

Solved Problem Example 2.3: Multi-Cohort Mathematical Forecasting of Population Aging and Aging Index

An actuary models the future aging trajectory of an urban state currently undergoing rapid fertility decline. At baseline ($t = 0$), the population is divided into three functional age cohorts:

Children ($0-14$ years): $P_0(C) = 1{,}800{,}000$
Working-Age Adults ($15-64$ years): $P_0(W) = 5{,}400{,}000$
Elderly ($65+$ years): $P_0(E) = 800{,}000$

Total baseline population is $N_0 = 8{,}000{,}000$.

Over the subsequent 25-year projection horizon ($t = 25$):

Sub-replacement fertility ($TFR = 1.2$) causes the incoming cohort of children to shrink at an annual exponential rate of $r_C = -0.018$ ($-1.8\%$ per year).
The working-age cohort transitions into late career, growing at a modest rate of $r_W = +0.004$ ($+0.4\%$ per year).
Longevity gains and the aging of large mid-century baby boom cohorts cause the elderly population ($65+$) to expand at a rapid rate of $r_E = +0.032$ ($+3.2\%$ per year).

Calculate the projected population in each of the three functional age groups at $t = 25$ years using the continuous compounding model $P_{25} = P_0 e^{r \cdot 25}$.
Determine the baseline ($t=0$) and projected ($t=25$) values for:

Child Dependency Ratio ($CDR$)
Old-Age Dependency Ratio ($ODR$)
Total Dependency Ratio ($TDR$)
Aging Index ($AI = P_E / P_C \times 100$)

Analyze the compositional transition and explain why the Total Dependency Ratio can experience a U-shaped trajectory during the aging process.

Project Cohort Sizes at $t = 25$ Years

Using the continuous compounding formula $P(25) = P_0 e^{25 r}$:

1. Children ($0-14$):

r_C \times 25 = -0.018 \times 25 = -0.450

e^{-0.450} = 0.637628

P_{25}(C) = 1{,}800{,}000 \times 0.637628 = 1{,}147{,}730

2. Working-Age Adults ($15-64$):

r_W \times 25 = 0.004 \times 25 = +0.100

e^{+0.100} = 1.105171

P_{25}(W) = 5{,}400{,}000 \times 1.105171 = 5{,}967{,}923

3. Elderly ($65+$):

r_E \times 25 = 0.032 \times 25 = +0.800

e^{+0.800} = 2.225541

P_{25}(E) = 800{,}000 \times 2.225541 = 1{,}780{,}433

Total Projected Population at $t = 25$:

N_{25} = 1{,}147{,}730 + 5{,}967{,}923 + 1{,}780{,}433 = 8{,}896{,}086

Calculate Baseline and Projected Demographic Metrics

Now, compute all ratios at both temporal horizons:

At Baseline ($t = 0$):

Child Dependency Ratio:

CDR_0 = \frac{1{,}800{,}000}{5{,}400{,}000} \times 100 = 33.33 \text{ per 100 workers}

Old-Age Dependency Ratio:

ODR_0 = \frac{800{,}000}{5{,}400{,}000} \times 100 = 14.81 \text{ per 100 workers}

Total Dependency Ratio:

TDR_0 = 33.33 + 14.81 = 48.15 \text{ per 100 workers}

Aging Index:

AI_0 = \frac{P_0(E)}{P_0(C)} \times 100 = \frac{800{,}000}{1{,}800{,}000} \times 100 = 44.44 \text{ seniors per 100 children}

At Projection Horizon ($t = 25$):

Child Dependency Ratio:

CDR_{25} = \frac{1{,}147{,}730}{5{,}967{,}923} \times 100 = 19.23 \text{ per 100 workers}

Old-Age Dependency Ratio:

ODR_{25} = \frac{1{,}780{,}433}{5{,}967{,}923} \times 100 = 29.83 \text{ per 100 workers}

Total Dependency Ratio:

TDR_{25} = 19.23 + 29.83 = 49.07 \text{ per 100 workers}

Aging Index:

AI_{25} = \frac{P_{25}(E)}{P_{25}(C)} \times 100 = \frac{1{,}780{,}433}{1{,}147{,}730} \times 100 = 155.13 \text{ seniors per 100 children}

Demographic Trajectory and the U-Shaped Dependency Dynamic

A superficial comparison of the Total Dependency Ratio might suggest that overall demographic dependency barely changed over the quarter-century:

TDR_0 = 48.15 \implies TDR_{25} = 49.07 \quad (\text{a negligible net increase of } 0.92)

However, examining the constituent components reveals a colossal structural metamorphosis:

1. The Aging Index Explosion: The Aging Index skyrocketed from $44.44$ to $155.13$—a massive $249\%$ surge! Whereas in 2000 there were more than two children for every senior citizen, by 2025 senior citizens outnumber children by over $55\%$.

2. The Substitution of Dependency: Child dependency plummeted from $33.33$ down to $19.23$ (due to low fertility), but this relief was entirely cancelled out by old-age dependency doubling from $14.81$ to $29.83$.

3. The U-Shaped Dependency Trajectory:

During the first 10–15 years of the projection, child dependency fell much faster than old-age dependency rose, driving $TDR$ downward to an absolute minimum of approximately $42$ (the peak of the demographic dividend). But as the baby-boom generation entered retirement after year 15, the relentless compounding of $r_E = +3.2\%$ caused $ODR$ to accelerate, driving $TDR$ back up in a pronounced U-shaped curve.

4. Fiscal Implications: In modern public finance, an elderly dependent is substantially more costly to state budgets than a child dependent (healthcare, pension annuities, and long-term care costs exceed primary school education costs by a factor of 3 to 5). Thus, an identical $TDR \approx 49$ represents a vastly more demanding fiscal environment when dominated by senior citizens rather than children.